1. Find the limit,
$\lim_{t \to \infty} x^t$
for $x \in [0, 1]$ Hint: It will depend on $x$.
2. Sketch the graph for a few values of $t$. Feel free to use Desmos.
3. If we define a function that depends on a parameter $t$, say $f_t(x) = x^t$, what
can we say about the limit? If $f_t(x)$ is continuous for different values of $t$ will
the $\lim_{t \to \infty} f_t(x)$ be continuous? Explain.
4. You may have seen from the previous problem that \"nice\" functions can do
strange things if we take a limit. Sometimes these things can be bad. However,
we have a remedy for these types of bad things, the integral. Integration is a
sort of \"smoothing\" process. It can \"overlook\" discontinuities, as long as the
set of discontinuities has measured zero. E.g. if you measure the interval $[5, 10]$
you would say it has length 5. But if I asked you what the length of 5 is, you
would probably say zero, since it is a point. Now, this is of interest because
the sets of measure zero for us will be points.
Consider
$g_t(x) = \frac{e^{tx}}{1 + e^{tx}}$
5. What is the $\lim_{t \to \infty} g_t(x)$? Hint: It will depend on $x$.
6. Use Desmos to graph $g_t(x)$ for different values of $t$. This will also help you
visualize the limit.
7. Okay, so the limit(s) of $g_t(x)$ are not so nice. Remember, we like smooth
functions. BUT, this function, and its limits, can still make sense under an
integral. What is
$\int_0^1 \lim_{t \to \infty} \frac{e^{tx}}{1 + e^{tx}} dx$?
Does it equal
$\lim_{t \to \infty} \int_0^1 \frac{e^{tx}}{1 + e^{tx}} dx$?
8. When can you move a limit in and out of an integral?
